Content uploaded by Oyvind Gron

Author content

All content in this area was uploaded by Oyvind Gron on Jul 10, 2014

Content may be subject to copyright.

INSTITUTE OF PHYSICS PUBLISHING EUROPEAN JOURNAL OF PHYSICS

Eur. J. Phys. 27 (2006) 885–889 doi:10.1088/0143-0807/27/4/019

The twin paradox in the theory of

relativity

ØGrøn

Oslo College, Department of Engineering, PO Box 4, St Olavs Pl., 0130 Oslo, Norway

and

Institute of Physics, University of Oslo, PO Box 1048 Blindern, 0216 Oslo, Norway

Received 9 January 2006

Published 30 May 2006

Online at stacks.iop.org/EJP/27/885

Abstract

It is pointed out that a complete resolution of the twin paradox demands that the

travelling twin takes into account the gravitational effect upon the rate of time

when he predicts the ageing of his brother. Two ways of making this prediction

are presented. The ﬁrst one is formally very simple. However, it involves an

assumption that is not obvious when the travelling twin stipulates the distance

of his brother. The second method makes use of Lagrangian dynamics in a

uniformly accelerated reference frame. Then no such assumption is necessary.

1. Introduction

There exist more than three hundred articles about the twin paradox [1–8]. In the present

paper, I shall present two ways that the travelling twin can predict the ageing of his brother. As

far as I have found, none of these calculations have been presented before. They are interesting

because they are simple and exact. In both calculations, one has to take into account the effect

of gravity upon time. Hence, a complete resolution of the twin paradox involves the general

theory of relativity.

This is not so surprising. The formulation of the twin paradox makes use of the general

principle of relativity, which is not contained in the special theory of relativity. One says:

One twin, say Eartha, at rest on the Earth, predicts that her sister, Stella, is younger than

herself when they meet again after Stella’s travel, because of the kinematic time dilation. But

according to the general principle of relativity, Stella can consider herself as at rest during all

of the travel, and would then predict that Eartha is younger than herself when they meet again.

The twin paradox is the conﬂict between these predictions.

Hence, the twin paradox arises from using general relativity in its formulation and only

special relativity to calculate the ageing that each twin predicts for the other. In the present

paper, it will be shown how general relativity is involved in the solution of the twin paradox.

0143-0807/06/040885+05$30.00 c

2006 IOP Publishing Ltd Printed in the UK 885

886 ØGrøn

2. Elementary solution of the twin paradox

The star Alpha Proxima is L0=4 ly (light years) from the Sun. Stella travels with constant

velocity v=0.8cto Alpha Proxima and back. To simplify the calculation we assume

that the acceleration at the start, the turning point and the arrival is so large that these

brief periods of acceleration can be neglected. As measured by Eartha the travel time is

t =2L0/v =8ly/0.8c=10 years. Hence, Eartha predicts that she will be 10 years older

when she meets Stella after her travel, and that Stella will only be

τS=1−v2/c2(2L0/v) =1−(0.8c)2/c2·10 years =6 years (2.1)

older. According to Eartha, Stella has become 4 years younger than herself during the travel

due to the velocity-dependent time dilation.

Let us now see what Stella predicts. The general principle of relativity says that she

can consider herself as at rest and Eartha as the traveller during all of the time when they

are separated from each other. She observes that the Earth and Alpha Proxima travel with

the velocity v=0.8c. Then the distance between the Earth and Alpha Proxima is Lorentz

contracted, so the distance is L=L01−v2/c2=4ly0.6=2.4 ly. Stella calculates that

the time Eartha uses forth and back is 2(L/v) =2(2.4/0.8)years =6 years. So Stella

predicts that she ages by 6 years during Eartha’s travel. This agrees with Eartha’s prediction.

Stella’s prediction of Eartha’s ageing remains. Due to the velocity-dependent time dilation

she ﬁnds that Eartha ages by 1−v2/c2(2L/v) =0.6·6 years =3.6 years during her forth

and back travel, in contradiction to Eartha’s own prediction that she ages by 10 years. This

disagreement is just the twin paradox.

What has not yet been taken into account is the effect upon time of the gravitational ﬁeld

that Stella experiences at the start, the turning point and the arrival. We assume that Stella

has constant proper acceleration, g, i.e. constant acceleration relative to her instantaneous

inertial rest frame, during these periods. Hence, she may be considered at rest in a uniformly

accelerated reference frame. In such a frame the line-element has the form

ds2=−

1+ gx

c22

c2dt2+dx2+dy2+dz2.(2.2)

The general physical interpretation of the line element in the case of a timelike spacetime

interval is that it represents the proper time interval dτas measured by a clock following a

worldline connecting two nearby events with coordinate separations (dt,dx, dy, dz), i.e.

ds2=−c2dτ2.(2.3)

For a clock at rest at the position x=hthis gives

τ =1+ gh

c2t. (2.4)

Stella chooses the coordinate system so that she is at the origin each time she accelerates,

i.e. each time she experiences a gravitational ﬁeld. At the start and the arrival Eartha is at

a small distance from her. In the limit of inﬁnite acceleration corresponding to vanishingly

short periods of acceleration, the periods with gravity at the start and arrival can be neglected.

Consider the start, for example. The fact that gh is ﬁnite in the limit g→∞is most

simply seen from the Galilean equation v2=2gh since the velocity is ﬁnite. The relativistic

formulae are more complicated, but lead to the same result. Furthermore tstart =v/g

vanishes as g→∞. Hence τstart vanishes at the start in this limit, and likewise at the arrival.

Note that this limit must be taken by Stella in order that she shall consider the same situation

The twin paradox in the theory of relativity 887

as Eartha, who reckoned with constant velocities all the way, neglecting the periods when

Stella had accelerated motion.

Let us now see how Stella can predict the ageing of Eartha when she experiences a

gravitational ﬁeld at the turning point close to Alpha Proxima. The Earth with Eartha falls

freely upwards in the gravitational ﬁeld she experiences. When the Earth stops and begins

to fall down it is at a height h=4 ly. Stella experiences the gravitational ﬁeld for a time

t =2v/g. Hence, Stella calculates that Eartha ages by

τ =1+ gh

c2t =1+ gh

c22v

g

=

2v

g+2hv

c2.(2.5)

The ﬁrst term vanishes in the limit g→∞. In this limit, Stella ﬁnds that Eartha ages by

τ =

2hv

c2=

2·4ly0.8c

c2=6.4 years (2.6)

during the time she changes from moving freely upwards with velocity vto falling downwards

with the same velocity.

Stella found that Eartha aged by 3.6 years during the forth and back travel. Hence the total

ageing of Eartha according to Stella is 10 years in agreement with Eartha’s own prediction.

3. Using relativistic Lagrangian dynamics in solving the twin paradox

We shall now see how Stella can calculate the ageing of Eartha during the time she experiences

a gravitational ﬁeld when she is close to Alpha Proxima, without making the assumption that

the distance of Eartha from her can be put equal to 4 ly during the whole of this period.

Eartha is moving freely in the gravitational ﬁeld experienced by Stella. The covariant

Lagrangian of Eartha, considered as a free particle with unit mass, is

L=1

2gµν ˙

xµ˙

xν(3.1)

where a dot denotes differentiation with respect to Eartha’s proper time. In the present case,

this leads to

L=−

1

21+ gx

c22

c2˙

t2+1

2˙

x2.(3.2)

Since the Lagrangian is independent of the time, the canonical momentum

pt=−

1+ gx

c22

c˙

t(3.3)

is a constant of motion. The four-velocity identity takes the form

−1+ gx

c22

c2˙

t2+˙

x2=−c2.(3.4)

Inserting the expression for ptinto the four-velocity identity gives

p2

t−c21+ gx

c22

=1+ gx

c22˙

x2(3.5)

or

dτ=

1+ gx

c2

p2

t−c21+ gx

c22dx. (3.6)

888 ØGrøn

Eartha’s turning point is at x=h. Hence, the value of ptmay be determined from the

condition ˙

x=0forx=h. Using equation (3.4)thisgives

pt=c1+ gh

c2.(3.7)

Let x1be Eartha’s position in Stella’s coordinate system at the moment when Stella ﬁrst

experiences the gravitational ﬁeld. At this moment the Earth and Alpha Proxima move upwards

with a velocity vand a Lorentz contracted distance, x1=2.4 ly. During the slowing down

the distance of Eartha increases from x1=2.4lyto x2=h=4 ly. Integrating equation (3.5)

from x1to x2and inserting the value of pt, Stella ﬁnds that during this slowing down Eartha

ages by

τ1−2=

c

g1+ gx2

2

c22

−1+ gx2

1

c22

.(3.8)

Taking the limit with inﬁnite proper acceleration, Stella obtains

lim

g→∞

τ1−2=

1

cx2

2−x2

1.(3.9)

Inserting x2=4 ly and x1=2.4 ly. Stella ﬁnds limg→∞ τ1−2=3.2 years. So Stella ﬁnds that

Eartha’s ageing as she changes her motion from upwards with velocity v=0.8cto downwards

with the same velocity is τEartha =2 limg→∞ τ1−2=6.4 years. Hence, Stella predicts a total

ageing of Eartha, 3.6 years + 6.4 years =10 years, again in agreement with the prediction

of Eartha herself.

4. Conclusion

The formulation of the twin paradox involves the general theory of relativity by making use

of the general principle of relativity. It arises when the ‘travelling’ twin considers herself as at

rest during all parts of the travel, including the periods when she is accelerated, and predicts

that the Earthbound twin is younger than herself when they meet after the departure, making

use of the special relativistic kinematic time dilation, only.

Its solution is to recognize that the travelling twin must take the effect of gravity upon

time into account when she calculates the age of her sister. When the ‘travelling’ twin turns

around at Alpha Proxima she experiences a gravitational ﬁeld directed away from her sister.

Hence, the Earthbound sister is higher up in this ﬁeld than herself, and ages faster. This effect

is greater the stronger the gravitational ﬁeld she experiences, so it cannot be neglected even in

the limit of a vanishing short period of acceleration, because the proper acceleration increases

towards inﬁnity in this limit.

Although Eartha and Stella both consider themselves as at rest, they both predict that

Eartha ages by 10 years during their departure and Stella only 6 years. But they have totally

different explanations for this difference of age when they meet again. Eartha says that Stella

is younger than herself because of Stella’s slower ageing due to her great velocity. Stella,

on the other hand, says that Eartha is older than herself because she ages so fast when Stella

experiences a gravitational ﬁeld close to Alpha Proxima. This illustrates a general property of

physical theories. The explanation that an observer gives of a phenomenon depends upon the

motion of the observer.

The twin paradox in the theory of relativity 889

References

[1] Marder L 1971 Time and the Space-Traveller (London: George Allen & Unwin)

[2] Møller C 1972 The Theory of Relativity 2nd edn (Oxford: Oxford University Press) section 8.17

[3] Fock V 1966 The Theory of Space, Time and Gravitation 2nd edn (New York: Pergamon) section 62

[4] Unruh W G 1981 Parallax distance, time, and the twin ‘paradox’ Am.J.Phys.49 589–92

[5] Eriksen E and Grøn Ø 1990 Relativistic dynamics in uniformly accelerated reference frames with application to

the clock paradox Eur. J. Phys. 11 39–44

[6] Debs T A and Redhead M L G 1995 The twin ‘paradox’ and the conventionality of simultaneity Am. J. Phys.

64 384–92

[7] Nicoli´

c H 2000 The role of acceleration and locality in the twin paradox Found. Phys. Lett. 13 595–601

[8] Iorio L 2006 An analytical treatment of the Clock Paradox in the framework of the special and general theories

of relativity Found. Phys. Lett. 18 1–19